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Lead Coefficients of Completing the Square

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

Quadratic equations are used in many areas. This lesson will teach you how to solve them by completing the square, using real-world examples where the lead coefficient is not one.

Solving Quadratics by Completing the Square

Have you ever heard the expression, 'It's not rocket science?' when it comes to learning mathematics? Well in the case of quadratic equations, it actually is rocket science! The path a rocket takes when it is launched is in the shape of a parabola, which is the graph of a quadratic equation. The rocket is launched up, then it slows, turns, and falls back to the ground, making the shape of an upside down 'U.' This path that a rocket takes can be modeled by a quadratic equation. A quadratic equation is an equation in which the highest exponent is two.


standard form


Since we can use quadratic equations to represent the path of rockets, we can use what we know about these equations to answer questions about the rocket. We can determine the maximum height of the rocket, the time at which it reaches its maximum, and at what time the rocket will land on the ground. To determine what time the rocket will land on the ground, we will use the method of completing the square.

Example 1:

The path of a rocket can be modeled by the equation below, where h is the height, in feet, of the rocket after t seconds. When will the rocket hit the ground?


example 1


If we are trying to determine when the rocket will hit the ground, we really want to know the time, t, when the height of the rocket is 0. Since we know that the height is zero, we are going to replace h(t) with 0.


time = 0


Since we now have one variable, t, to solve for, we will complete the square. There are six steps to completing the square. Taking a quadratic equation in standard form:


standard


Step 1: Move the constant to the other side of the equation. In other words, subtract c from both sides of the equation.

In this example, there is no constant term to move.

Step 2: Factor out the lead coefficient in order to make the new lead coefficient equal to 1. In other words, divide each term by a, placing the original lead coefficient outside the parentheses.


step 2


Step 3: To complete the square we must take the coefficient of the x, divide it by two, and square it, and then add it to both sides of the equation.


b over 2


To complete the square, we take the b value and divide it by two and then square it. -6 divided by 2 is -3 and (-3)(-3)=9. Therefore, we are going to add 9 to the right side of the equation.

We need to add the same amount to the left side of the equation to keep it equal. Since we divided out the -4 in step two, we need to multiply 9 by -4, which gives us -36 to add to the left side.


step 3


Step 4: Now we can write the left side of the equation as a perfect square.

Since (-3)+(-3)= -6 and (-3)(-3)= 9, we can write it as a prefect square.


step 4


Step 5: Solve the equation for the variable.

We will solve this equation for t by dividing both sides by -4, then taking the square root of both sides, then adding 3 to both sides.


5a


5b


5c


Step 6: Simplify to find your answers.

This equation will give us two answers because when you take the square root of a number, it can either be positive or negative.


answers


In this case, our answers are 0 seconds and 6 seconds. In terms of the problem, it makes sense that we get two answers. The rocket is on the ground at zero seconds because it hasn't been launched yet. The rocket is then launched, and it goes up and comes back down to the ground at 6 seconds. The answer we are looking for is 6 seconds.

Example 2:

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